Conversely, suppose there is a fan-crossing free drawing of \(G_I\) ,
where the drawing is on the sphere. As in [1]}, we
place \(c_T\)  at the North Pole and \(c_C\)  at the South Pole.
A meridian path \(M_i\)  is a path of five fat edges between
the centers of the transmitter and the collector through the
vertices \(u_{3i}\)  and \(v_{Bi}\)  on the boundaries, for \(i=1,\ldots ,m\) . Each meridian path also contains a unique path of length five
from \(c_T\)  to \(c_C\) . A splitter path \(SP_a = (c_T, a_T,a^{\prime }_T,c_a, a^{\prime }_C, a_C, c_C)\)  is a path of length six between the
centers of the transmitter and the collector via the center of a
splitter \(P_a\) , such that \(a_T\)  and \(a_C\)  are satellites of \(P_a\) 
and \(a^{\prime }_T\)  and \(a^{\prime }_C\)  are vertices between \(a_T\)  and \(c_a\)  and \(c_a\) 
and \(c_C\) , respectively. Note that there is a splitter path from
\(c_C\)  through \(a_C\)  and \(c_a\)  for \(a\)  vertices and there is a single
path from \(c_T\)  through \(a_T\)  to \(c_a\) .
